Precalculus Examples

$y - 2 = 2 {(x + 1)}^{2}$

Step 1

Find the x-intercepts.

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To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$(0) - 2 = 2 {(x + 1)}^{2}$

Solve the equation.

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Rewrite the equation as $2 {(x + 1)}^{2} = (0) - 2$ .

$2 {(x + 1)}^{2} = (0) - 2$

Subtract $2$ from $0$ .

$2 {(x + 1)}^{2} = - 2$

Divide each term in $2 {(x + 1)}^{2} = - 2$ by $2$ and simplify.

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Divide each term in $2 {(x + 1)}^{2} = - 2$ by $2$ .

$\frac{2 {(x + 1)}^{2}}{2} = \frac{- 2}{2}$

Simplify the left side.

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Cancel the common factor of $2$ .

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Cancel the common factor.

$\frac{2 {(x + 1)}^{2}}{2} = \frac{- 2}{2}$

Divide ${(x + 1)}^{2}$ by $1$ .

${(x + 1)}^{2} = \frac{- 2}{2}$

Simplify the right side.

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Divide $- 2$ by $2$ .

${(x + 1)}^{2} = - 1$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$x + 1 = \pm \sqrt{- 1}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x + 1 = \pm i$

The complete solution is the result of both the positive and negative portions of the solution.

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First, use the positive value of the $\pm$ to find the first solution.

$x + 1 = i$

Move all terms not containing $x$ to the right side of the equation.

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Subtract $1$ from both sides of the equation.

$x = i - 1$

Reorder $i$ and $- 1$ .

$x = - 1 + i$

Next, use the negative value of the $\pm$ to find the second solution.

$x + 1 = - i$

Move all terms not containing $x$ to the right side of the equation.

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Subtract $1$ from both sides of the equation.

$x = - i - 1$

Reorder $- i$ and $- 1$ .

$x = - 1 - i$

The complete solution is the result of both the positive and negative portions of the solution.

$x = - 1 + i, - 1 - i$

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

x-intercept(s): $None$

Step 2

Find the y-intercepts.

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To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y - 2 = 2 {((0) + 1)}^{2}$

Solve the equation.

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Simplify $2 {((0) + 1)}^{2}$ .

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Add $0$ and $1$ .

$y - 2 = 2 \cdot 1^{2}$

One to any power is one.

$y - 2 = 2 \cdot 1$

Multiply $2$ by $1$ .

$y - 2 = 2$

Move all terms not containing $y$ to the right side of the equation.

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Add $2$ to both sides of the equation.

$y = 2 + 2$

Add $2$ and $2$ .

$y = 4$

y-intercept(s) in point form.

y-intercept(s): $(0, 4)$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, 4)$

Step 4